Magnetized peristaltic particle–fluid propulsion with Hall and ion slip effects through a permeable channel

Journal Pre-proof Magnetized peristaltic particle-fluid propulsion with Hall and ion slip effects through a permeable channel A. Zeeshan, M.M. Bhatti, Taseer Muhammad, Lijun Zhang

PII: DOI: Reference:

S0378-4371(19)32214-9 https://doi.org/10.1016/j.physa.2019.123999 PHYSA 123999

To appear in:

Physica A

Received date : 29 April 2019 Revised date : 12 October 2019 Please cite this article as: A. Zeeshan, M.M. Bhatti, T. Muhammad et al., Magnetized peristaltic particle-fluid propulsion with Hall and ion slip effects through a permeable channel, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123999. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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*Highlights (for review)

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Peristaltic transport of particle-fluid suspension through a permeable channel is investigated. Flow and heat transfer properties are considered.



Hall and ion slip effects are taken into account.



Exact solutions are developed by software MATHEMATICA.

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*Manuscript Click here to view linked References

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Magnetized peristaltic particle-fluid propulsion with Hall and ion slip effects through a permeable channel A. Zeeshana, M.M. Bhattib, Taseer Muhammadc*, Lijun Zhangb a

Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan College of Mathematics and Systems Science, Shandong University of Science and

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b

Technology, Qingdao, Shandong 266590, China c

Department of Mathematics, Government College Women University, Sialkot 51310, *

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Pakistan

Corresponding author E-mail: [email protected] (Taseer Muhammad)

Abstract: In this article, combine impacts of the Hall and ion slip with heat exchange on the peristaltic movement of MHD particle-fluid suspension through a permeable channel have

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been examined. The representing flow issue for the liquid stage and dusty stage have been displayed with the assistance of long wavelength and disregarding the inertial forces. Correct arrangements are gotten for velocity and temperature profile. The effect of all the applicable parameters, for example, particle volume fraction, Hartmann number, ion slip parameter, Hall parameter, Darcy number, Prandtl number, Eckert number, are portrayed for velocity and temperature profile. It is observed that the velocity profile appears inverse close to the

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qualities for bigger estimations of the Darcy parameter, however, it diminishes because of Hartmann number and particle volume fraction. Moreover, the behaviour of velocity profile stays comparative for hall and ion slip parameter. The Darcy parameter also shows significant

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resistance to the temperature profile.

Keywords: Heat transfer; Permeable channel; Particle-fluid; Ohm’s Law; Hall effect.

1. Introduction

Over the last few decades, non-Newtonian fluid applications have developed in many

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industrial processes such as catalytic chemistry, biology, medicine, and environmental applications [1-5]. As of late, the investigation of non-Newtonian liquid models has expanded massively because of the expanding significance of non-Newtonian liquids in inventive innovation and current enterprises. For planning ideal stream design and for choosing conditions, displaying of non-Newtonian streams is essential to foresee, and by along these lines, one can anticipate and improve understanding the conduct of procedures. Cases of nonNewtonian liquid models include restorative items, blood, body-liquids and oil, and so forth. 1

Journal Pre-proof In the organic sciences, we can watch that the impossible to miss worm-like wave movement of the digestion tracts and other empty solid structures, delivered by the progressive compression of the strong filaments of their dividers, constraining their substance ahead, this particular way the movement of warm is known as a peristaltic movement. The word peristaltic originates from the Greek word peristaltikos, which means grasping and compacting. This movement can likewise be seen in the development of eggs in a Fallopian

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tube, the vehicle of spermatozoa in the cervical waterway, transport of cilia, flow in little veins, and working of a ureter and numerous others. The essential reasons for the peristaltic

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movement are to break down smooth movement inside the pump and the smooth auxiliary movement as a conceivable liquid blending system. Latham [6] initiated to investigate human ureter, which pumps fluid by mean of peristaltic motion. After that, numerous theoretical attempts have been made by several researchers and developed a variety of fluid models for peristaltic flows [7-10].

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Peristaltic course with molecule fluid gives been considered by different specialists. For instance, Mekheimer et al. [11] considered the peristaltic activity of molecule fluid suspension through a planar channel. Nagarani and Sarojamma [12] examined the peristaltic movement of little contaminants utilizing the power-law fluid unit through a channel. Mekheimer et al. [13] dissected the particulate suspension stream in light of the sinusoidal peristaltic wave employing an unconventional chamber. Of late, Mekheimer and Mohamed

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[14] inspected the peristaltic activity of the pulsatile flow of molecule fluid suspension through an annular area utilizing a blood coagulum display. Significantly more scientific and numerical examines on peristaltic stream, and liquid molecule suspensions are accessible

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from refs. [15-20]. Peristaltic development with temperature and mass exchange additionally has numerous business applications, for instance, convection of high temperature in blood course, conduction in tissues, foodstuff handling, vasodilation, hemodialysis, oxygenation, switch osmosis, ignition refining, and process methodology and so forth. A couple of

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specialists has just researched peristaltic flow with mass and temperature exchange. For example, Mekheimer et al. [21] considered the results of warming and mass exchange on peristaltic development in a vertical permeable channel. Hayat et al. [22] researched the outcomes of warmth and mass exchange on MHD peristaltic development of non-Newtonian Maxwell fluid having consistent dividers. By and by, Hayat et al. [23] examined the impact of divider impacts with temperature and mass exchange on a peristaltic stream of third-grade fluid in a bended channel. Ellahi et al. [24] dissected the peristaltic development of temperature and mass exchange of thick fluid in a non-uniform rectangular channel. 2

Journal Pre-proof Magnetohydrodynamics (MHD) plays an essential role and has remarkable importance in conducting various physiological materials such as blood pumps, blood, and magnetohydrodynamics compressor. Magnetohydrodynamics is also applicable and helpful in biomedical engineerings such as Magnetic drug targeting and magnetic resonance imaging etc. Besides, the impact of hall and particle slip has gotten an impressive consideration in MHD streams in which electromagnetic power is noticeable. Consolidate consequences of the

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hall and particle slip with warm exchange processes are found in different building applications such as power generators, refrigeration loops, corridor quickening agents, MHD

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quickening agents, warm components, and electric transformers. In addition, the investigation of hall present and attractive on the bloodstream is especially useful to make sense of the attractive reverberation angiography, which is helpful to take pictures of corridors to comprehend and imagine them for stenosis or distinctive variations from the norm. Specifically, the conduits of mind and neck, the stomach aorta, the thoracic, and the renal

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courses. Different creators researched MHD stream issues with consolidated impacts of hall and particle slip in various geometrical angles. For example, Ellahi et al. [25] investigated the impact of the Hall and ion slip on the peristaltic stream of Jeffrey fluid through a non-uniform duct. Mekheimer [26] examined the peristaltic movement of magnetohydrodynamic (MHD) with the slip stream initiated by the surface acoustic wavy divider through a porous medium. Nowar [27] contemplated the Hall current on peristaltic nanofluid move through a permeable

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medium. Abo-Eldahab et al. [28] inspected the Hall and particle slip consequences for the peristaltic stream affected by magnetohydrodynamics (MHD). Some more germane investigations on the present point can be found from refs. [29-32] and several therein.

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With the above examination, the fundamental motivation behind a present examination is to inspect the synchronous impact of the Hall and particle slip with a warm exchange on the peristaltic movement of MHD molecule liquid suspension through a penetrable channel. The administering stream conditions for the liquid and particulate stage

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are displayed by taking the estimation of long wavelength and crawling stream administration. Correct arrangements are achieved for speed and temperature profile though the articulation for weight surge is assessed numerically with the assistance of computational codes. This paper is sorted out as, after the presentation in Sec. (1 )Sec. (2) delineates the plan of the issue, Sec. (3) portrays the arrangement approach, and Sec (4) at long last is spent significant time in numerical outcomes and exchange.

3

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velocity components Cartesian coordinate Particle Volume fraction pressure in fixed frame wave amplitude radius of the particles width of the channel wave velocity Eckert number Electric field Magnetic field current density Prandtl number Reynolds number Time Hartmann number Drag force Darcy number Thermal conductivity Volume flow rate Temperature and concentration coefficient of mass diffusivity Mean temperature Thermal diffusion ratio Specific heat at constant volume Greek symbols

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Dimensionless temperature fluid viscosity Electric conductivity of the fluid Viscosity of the fluid Wavelength Slip parameter Ion slip parameter Hall parameter Slip parameter Amplitude ratio Thermal equilibrium time Fluid density Subscripts Fluid phase Particulate phase

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2. Mathematical formulation Let us consider the sinusoidal motion of particles in a dusty viscous fluid through a twodimensional permeable channel of uniform thickness. The fluid is electrically conduction, and an external magnetic field is applied to it, while the induced magnetic field is considered to be zero here. We have selected the Cartesian coordinates system by taking is perpendicular to the channel (see Fig. 1).

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the centre of the channel, and the

along

Fig. 1. Geometry of the problem. The geometry of the wall surface can be written as

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(1) The generalized form of ohm's law with Hall and ion-slip effect can be written as [25]

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(2)

Solving Eq. (2) we get

(4)

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and

(3)

The governing equation of linear momentum, continuity and energy equation in the fixed frame can be written as [33] (i)

Fluid Phase:

(5)

5

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(6)

Dusty Phase:

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(ii)

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(7)

(8)

(9) (10) (11) (12)

In the present analysis, we considered the small particles; therefore, the interaction among particles has been neglected. The particle volume fraction is constant. The mathematical expression for the drag coefficient can be written as

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(13) Eq. (13) represents the relation for the classical stokes drag for small particle Reynold’s

function

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number modified to the account for a finite particulate volume fractional with the help of as presented by Tam [34]. The empirical relation for the viscosity suspension

can be written as [35]

(14)

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Let us define the transformation variable from a fixed frame to a wave frame, we have (15)

Introducing the following non-dimensional quantities

(16) Using Eq. (15) and Eq. (16) in Eq. (5) to Eq. (12), and taking the assumption of long6

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and zero Reynolds number

approximation. The momentum

equation for fluid phase can be written as (after dropping the tilde) (17) where

. The energy equation for the fluid phase reads

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(18) For particulate phase, it can be written as

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,

(19) (20)

Their corresponding non-dimensional boundary conditions are

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3. Solution of the problem

(21)

After integrating twice, the exact solution for velocity profile in simplified form can be

(22) (23)

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written as

where

(24)

.

(25)

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The volume flow rate for the fluid phase and particulate phase is given by

The pressure gradient

(26) is obtained after solving Eq. (26), we have

The non-dimensional pressure rise

(27) is evaluated numerically by using the following

expression 7

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(28)

4. Results and discussion This section describes the graphical behaviour of various physical quantities that are involved in the present flow problem. Computational software Mathematica has been utilized in order to examine the novelties of pressure rise, temperature profile and velocity profile

parameter

, particle volume fraction

, Hartmann number

, Ion slip parameter

, Hall

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against Darcy parameter

, Prandtl number

and Eckert number

respectively.

,

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In a human body, various biological moves with the help of peristaltic pumping and it is constructive to analyze different kinds of diagnostic problems. Pumping helps to transport different types of fluids such as sensitive fluids, noxious fluids, corrosive fluids and sanitary fluids. For this purpose, Fig. (2) and Fig. (3) are sketched to discuss the pumping

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characteristics. Numerical integration has been used to evaluate the term pressure rise in Eq. (26). Fig. (2) elucidates pumping rate decreases in retrograde pumping region , when hall parameter

and ion slip parameter

free pumping region

whereas its behavior is opposite in

and free pumping region

. It is clear

from Fig. (3) that pumping rate decreases in retrograde pumping region with the rise in particle volume fraction opposite in the co-pumping region

, however the response of pumping rate is and free pumping region

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Moreover, the influence of Hartmann number

is opposite on pressure rise as

compared to the particle volume fraction. Fig. (4) to Fig. (6) describes the velocity curves for

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all the physical parameters of interest. It can be observed from Fig. (4) that Hall parameter and ion slip parameter

enhance the velocity profile. From Fig. (5), we can analyze

that velocity distribution shows opposite behavior due to the more significant impact of Darcy parameter

, whereas the velocity profile decreases when the Hartmann number

increases. In fact, it is due to the influence of the Lorentz force, which opposes the flow,

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and hence, the velocity profile tends to decrease. It can be noticed from Fig. (6) that when the particle volume fraction

increases, then the magnitude of the velocity profile decreases.

Fig. (7) to Fig. (8) are drawn for temperature profile. Fig. (7) shows the simultaneous effect of Prandtl number that Prandtl number

and Eckert number and Eckert number

be illustrated from Fig. (8) that Hall parameter

. It can be observed from this figure enhances the temperature profile. It can and ion slip parameter

diminish the temperature profile. Fig. (9) is plotted against Hartmann number 8

tend to and

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. It can be seen from Fig. (9) that with the increment in particle

, temperature profile diminishes, whereas the more significant influence

of the magnetic field enhances the temperature profile. Fig. (10) is prepared from Darcy parameter

and particle volume fraction

Darcy parameter

. It can be examined from this figure that

oppose the temperature distribution, and hence, the temperature

profile decreases. The present results depict various interesting behaviour that warrants

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further study on particle-fluid suspension with various biological fluids.

Table 1: Numerical comparison for different values of governing parameters against velocity

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and temperature profile.

1.33134

3.83710

1.46624

1

1.35948

3.58986

1.32132

0.4

1.31277

4.00025

1.56542

0.9 1.5 1.9 0.5 7 0.4

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0.5

1.39238

3.30086

1.16096

1.39857

3.24643

1.30717

1.30717

4.04951

1.59590 1.87206 14.0274 1.27299 1.65949

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0.6

Fig. 2. Pressure rise vs volume flow rate for various values of 9

and

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and

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Fig. 3. Pressure rise vs volume flow rate for various values of

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Fig. 4. Velocity distribution for various values of

10

and

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Fig. 5. Velocity distribution for various values of

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Fig. 6. Velocity distribution for various values of

11

and

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and

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Fig. 7. Temperature profile for various values of

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Fig. 8. Temperature profile for various values of

12

and

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and

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Fig. 9. Temperature profile for various values of

Fig. 10. Temperature profile for various values of

and

.

5. Conclusions

In this study, the peristaltic flow of MHD particle-fluid suspension with heat transfer has been investigated through a permeable channel. Combine effects of Hall and ion slip are

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taken into account. The solutions for the governing coupled partial differential equations are obtained analytically, and closed form solutions are presented for fluid phase and dusty phase. The major points for the present flow problem are as follows: 

Velocity profile decreases with the increment in the magnetic field and particle volume fraction.

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The velocity profile also behaves as an increasing function due to the greater influence of Hall and ion slip effect. 13

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The temperature profile increases with the increment in Eckert number and Prandtl number.

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When the Hall and ion slip parameter increases, the temperature profile diminishes.

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The temperature profile also diminishes due to the influence of the Darcy parameter and particle volume fraction.

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Journal Pre-proof *Declaration of Interest Statement

Dear Editor, Hope you are fine and doing well. We have no conflict of interest for this submission.

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Best regards!

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Taseer Muhammad